Dissipative structures

28 Jan 1997 13:34

Ilya Prigogine (NL) coined the phrase, as a name
for the patterns which self-organize in far-from-equilibrium
dissipative systems. He thinks they're unbelievably important, and
says so at great length in his books. Some of us physicists believe him; some
are skeptical; I am leaning towards skepticism.

But to explain. Dissipation inspires the wrath of the moralist and the envy
of most others; for the physicist, however, it is merely faintly depressing.
We call something dissipative if it looses energy to waste-heat. (Technically:
if volume in the phase space is not conserved.) The famous Second Law of
Thermodynamics amounts to saying that, if something is isolated from the rest
of the world, it will dissipate all the free energy it has. Equivalently, it
maximizes its entropy. Thermal equilibrium is the state of maximum entropy.

If something is (in a well-defined sense) near thermal equilibrium, one can
show that its behavior is governed by linear differential equations (hence the
name "linear thermodynamics" for the appropriate body of theory), and that
left to itself it will approach equilibrium exponentially (hence the somewhat
more common name "irreversible thermodynamics"). Here we are guided, not by
the entropy, but by "entropy production," the rate of increase in entropy.
Since, once we reach equilibrium, the entropy cannot increase (by definition),
the entropy production at equilibrium is zero, and the entropy production is
always decreasing (the "principle of minimum entropy production").

In general, however, things are not well-isolated from the rest of the
world. If energy arrives from the outside as quickly as it is dissipated, even
bodies in the linear regime can be kept away from equilibrium. (Hence various
Creationist arguments about the Second Law are worthless: neither living things
nor the Earth are well-isolated from the rest of the universe, as may be
observed every day at sunrise.) Thus dissipation, and why dissipative systems
are not necessarily dull as dish-water. So you can have structures in
dissipative systems, and there's no reason not to call them "dissipative
structures", though it's not obvious that there are many interesting
generalizations about them.

"Far-from-equilibrium" means that your system is so far from its thermal
equilibrium that the linear laws I mentioned a moment ago no longer apply;
non-linear terms become important. The only general rule about the solution to
non-linear differential equations is that there are no general rules; hence the
interest in the subject. (Cf. Chaos and non-linear
dynamics.) This is not good news, of course, if what you want to do is
extend thermodynamics to the far-from-equilibrium case. But, one might
suppose, matters are not totally hopeless; we aren't talking about just any
arbitrary system of equations, but the particular ones important in
thermodynamics; perhaps there is some general principle (like those of maximum
entropy, or minimum entropy production) which can guide us to solutions. What
Prigogine claims to have done is to have found, if not another extremum
principle, then at least an inequality (a "universal evolution criterion"),
and to have used it to work out the theory of dissipative structures, according
to which patterns are supposed to form when the uniform, uninteresting
"thermodynamic branch" of the system becomes unstable. The math for all this
is analogous to that of equilibrium phase transitions with "broken symmetry",
where, again, a uniform state becomes unstable, forcing the system into a
patterned, coherent one to minimize free energy. Even without Prigogine's
claims that this theory is Very Significant to biology and social science, even
without the philosophical and cultural importance he claims for it, this would
be very interesting, and the big question is whether he's right, i.e., whether
and to what the theory applies, whether, so to speak, there are Dissipative
Structures and not just dissipative structures.

"Of course he's right," one is tempted to say. "Everyone acknolwedges
he's an expert on thermodynamics; he was part of the Brussels School which
basically invented irreversible thermodynamics; he won the Nobel Prize, for
crying out loud!" But irreversible thermodynamics is very different, and that
was a long time ago --- the forties and fifties and early sixties; that was
what the Nobel was for. ("And besides the wench is dead.")

And then there is the matter of his scientific peers --- not the systems
theorists and similar riff-raff, but the experts in thermodynamics and
statistical mechanics and pattern formation. One of them (P. Hohenberg,
co-author of the latest Review of Modern Physics book on the state
of the art on pattern formation) was willing to be quoted by Scientific
American (May 1995, "From Complexity to Perplexity") to the effect
that "I don't know of a single phenomenon his theory has explained."

This is extreme, but it becomes more plausible the more one looks into the
actual experimental literature. For instance, chemical oscillations and waves
are supposed to be particularly good Dissipative Structures; Prigogine and his
collaborators have devoted hundreds if not thousands of pages to their
analysis, with a special devotion to the Belousov-Zhabotisnky
reagent, which is the classic chemical oscillator. Unfortunately,
as Arthur Winfree points out (When Time Breaks Down, Princeton UP,
1987, pp. 189--90), "the Belousov-Zhabotinsky reagent ... is perfectly stable
in its uniform quiescence," but can be distrubed into oscillation and
wave-formation. This is precisely what cannot be true, if the theory
of Dissipative Structures is to apply, and Winfree accordingly judges that
"the first step [in understanding these phenomena], which no theorist would
have anticipated, is to set aside the mathematical literature" produced by a
"ponderous industry of theoretical elaboration". --- Needless to say,
Winfree is not opposed to theory or mathematics, and his superb The
Geometry of Biological Time (Springer-Verlag, 1980) is full of both.

Somewhat more diplomatic is Philip W. Anderson, one of the Old Turks of the
Santa Fe Institute, and himself a
Nobelist. I refer in particular to the very interesting paper he co-authored
with Daniel L. Stein, "Broken Symmetry, Emergent Properties, Disspiative
Structures, Life: Are They Related", in F. Eugene Yates (ed.),
Self-Organizing Systems: The Emergence of Order (NY: Plenum Press,
1987), p. 445--457. The editor's abstract is as follows:

The authors compare symmetry-breaking in thermodynamic
equilibrium systems (leading to phase change) and in systems far from
equilibrium (leading to dissipative structures). They conclude thgat the only
similarity between the two is their ability to lead to the emergent property of
spatial variation from a homogeneous background. There is a well-developed
theory for the equilbirium case involving the order parameter concept, which
leads to a strong correlation of the order parameter over macroscopic distances
in the broken symmetry phase (as exists, for example, in a ferromagnetic
domain). This correlation endows the structure with a self-scaled stability,
rigidity, autonomy or permanence. In contrast, the authors assert that there
is no developed thoery of dissipative structures (despite claims to the
contrary) and that perhaps there are no stable dissipative structures at all!
Symmetry-breaking effects such as vortices and convection cells in fluids ---
effects that result from dynamic instability bifurcations --- are considered to
be unstable and transitory, rather than stable dissipative structures.

Thus, the authors do not believe that speculation about dissipative
structures and their broken symmetries can, at present, be relevant to
questions of the origin and persistence of life.

Some quotes from the paper itself:

"Is there a theory of dissipative structures
comparable to that of equilibrium structures, explaining the existence of new,
stable properties and entities in such systems?"

Contrary to statements in a number of books and articles in this field,
we believe that there is no such theory, and it even may be that there
are no such structures as they are implied to exist by Prigogine, Haken, and
their collaborators. What does exist in this field is rather different from
Prigogine's speculations and is the subject of intense experimental and
theoretical investigation at this time.... [p. 447]

Prigogine and his school have made a series of attempts to build an
analogy between these [dissipative far-from-equilibrium systems which form
patterns] and the Landau free energy and its dependence on the order parameter,
which leads to the important properties of equilibrium broken symmetry systems.
The attempt is to generalize the principle of maximum entropy production, which
holds near equilibrium in steady-state dissipative systems, and to find some
kind of dissipation function whose extremum determines the state. As far as we
can see, in the few cases in which this idea can be given concrete meaning, it
is simply incorrect. In any case, it is clearly out of context in relation to
the observed chaotic behvaior of real dissipative
systems. [pp.454--455]

Anderson and Stein cite two of their own papers (P. W. Anderson, "Can
broken symmetry occur in driven systems?" in G. Nicolis, G. Dewel and
P. Turner (eds.), Order and Fluctuations in Equilbirium and
Non-Equilibirum Statistical Mechanics, pp. 289-297; and D. L. Stein,
"Dissipative structures, broken symmetry, and the theory of equilibrium phase
transitions," J. Chem. Phys.72:2869-2874) for
the technical details of their critique; I haven't read 'em yet. Their joint
paper is reproduced in Anderson's Basic Notions of Condensed Matter
Physics, sans illustrations. Prigogine may be observed waxing
philosophical in Order Out of Chaos, obscure in From Being
to Becoming, and textbookish in Self-Organization in
Non-Equilibrium Systems.